Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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1answer
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theorems that depend on the embedding of an affine variety into the affine space

Let $\mathcal{T}$ be a theorem regarding an affine variety $Y$ of $\mathbb{A}^n$. Question 1: What does the phrase "$\mathcal{T}$ does not depend on the embedding of $Y$ in $\mathbb{A}^n$" mean? ...
3
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1answer
76 views

Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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2answers
46 views

If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ “have the same distribution”?

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution? In a way, this seems correct: both $X$ and $-X$ have the same probability density ...
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0answers
57 views

Name of an aglebraic structures $(A,*,\cdot)$ weaker than semirings.

I have a set $A$ with two binary operations on it $(A,*,\cdot)$ STRUCTURE A $(A,*)$ is not commutative, is not associative, it has not an identity $(A,\cdot)$ is a commutative group $(a*b)\cdot ...
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2answers
104 views

LambertW: $ x=W(x\cdot e^{x}) $ for $ x \ge -1$ but not for $x \lt-1$. How do I express my formula/my text?

I just found by numerical heuristics for some systematic numbers $q(x)_\text{heuristical}$ depending on $x=1,2,3,4,\ldots$ using WolframAlpha the suggested interpretation in terms of the ...
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1answer
41 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
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3answers
53 views

What's the name of the set of products of equal to a given value?

Suppose we have the * operator on a set $A$ such that * is associative but not commutative. Given $a$, $b$, $c \in A$, \begin{align*} abc &= (abc) \\ &= (a)(bc) \\ &= (ab)(c) \\ &= ...
2
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3answers
136 views

Are the pre-image and the domain the same, or not?

Throughout school I thought that the pre-image was a subset of the domain, not that they were necessarily the same. When I spoke of a function f:R->R, I didn't think that this meant that f was defined ...
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6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...
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1answer
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When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
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1answer
87 views

What is the name of a graph made of k copies of a 4-cycle connected end to end in a chain, possibly with leaves?

Do graphs of the following sort have a specific name? We've been calling them Cactapillars, as they're cacti that look a little like caterpillars (and the name Caterpillar already refers to a ...
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2answers
60 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
1
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0answers
21 views

Enunciation of $\partial$ as the boundary map

How is $\partial$ typically pronounced when it is used as the boundary map in homology theory? The answer to this question provides some good information on the enunciation of $\partial$, but more ...
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3answers
46 views

Terminology: geometric sequences and geometric means

(I'll post my own answer to this one, but that should not deter others, since my answer is a surmisal.) Why are geometric sequences called geometric sequences? Whare are geometric means called ...
2
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3answers
143 views

Name the property $f(x) \ge x$

It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property: $$f(x) \ge ...
2
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0answers
25 views

what does “modular” mean?

I find some similarity of the concept "modular set functions" to the cardinality function. But I don't see the cardinality function is also called "modular" or something else. I wonder what "modular" ...
5
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0answers
27 views

“Advective”, “diffusive”, “dispersive”, and related terms in the realm of PDEs

Whenever I read a paper involving PDEs, the discussion inevitably refers to “the dispersive term” or “the advective term” or similar. From context it is usually possible to figure out the antecedent, ...
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0answers
20 views

What does “modular” in “modulr functions” mean?

From Wikipedia If $\Omega$ is a set, a submodular function is a set function $f:2^{\Omega}\rightarrow \mathbb{R}$, where $2^\Omega$ denotes the power set of $\Omega$, which satisfies one of the ...
2
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3answers
242 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
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2answers
44 views

Complete vs Perfect infomation in Combinatorial game theory

In their book "Winning Ways for Your Mathematical Plays", Berlekamp, Conway, and Guy used as the 7th condition for a combinatorial game "Both players know what is going on; There is complete ...
0
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0answers
23 views

Terminology in universal algbera

(Fix throughout a functional language $\Sigma$.) Given an algebra $A$ with underlying set $\vert A\vert$, there is an obvious surjective homomorphism from $A$ to the free algebra generated by $\vert ...
2
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3answers
60 views

Standard terminology for infinite limits with opposite sign on the two sides?

Consider the following limits: $$ \lim_{x\rightarrow0}\frac{1}{x^2}$$ $$ \lim_{x\rightarrow0}\frac{1}{x}$$ As far as I can tell, most authors say as a matter of terminology that these limits don't ...
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0answers
32 views

Addition, multiplication, exponentiation… What is next function of this series?

Addition can be (informally) defined as the application of successor function $S$ on $a$ $b$ times, i.e. $a+b=S\stackrel{b}{\cdots}S a$. Multiplication can be defined as the addition of $a$ with ...
6
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1answer
52 views

What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
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0answers
40 views

A term for category where every loop of morphisms is an identity

"A category where composition of every loop of morphisms is an identity." Moreover, in the case I am thinking about, morphisms are bijective functions. Is there a name for this concept?
0
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1answer
39 views

Help to conceive a name

Filter $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ Ideal $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I ...
1
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1answer
27 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
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2answers
90 views

Why do some sources call calculus, “the calculus”?

No need to cite specific sources since I think it's a fairly common thing to see. What's up with that? Thank you Edit: I've seen it in several places. Here's where I'm currently looking at it at: ...
2
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0answers
97 views

Alternative to sin and cos

I was reading something on the Internet the other day, and I swear I came across a reference to an alternative sine function [which I now cannot find any mention of]. The usual sine function starts ...
5
votes
1answer
114 views

Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
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2answers
40 views

What is the difference of an n-tuple and a permutation of n elements

My understanding of n-tuple and a permutation of n elements is, that both are ordered sequences of n elements. Are there differences in the objects correlating to these two terms ? I guess it ...
0
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1answer
44 views

Writing a chain of implications in English

How to write a theorem of the form $A\Rightarrow B\Rightarrow C\Rightarrow D$ where every $A$, $B$, $C$, $D$ are formulated with words (English) rather than with formulas? One idea: The next item of ...
2
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2answers
56 views

What is meant by a “structure map”?

The title is the question. Somehow I should know the answer, but I am by no means sure what is meant exactly by it. Perhaps it doesn't have a definite meaning and only in context, could someone ...
4
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0answers
46 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
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0answers
15 views

What is the sigmoid *squashing* function?

I've just read the following The basic unit ("neuron" i) performs the following computation to update its state $y_i$: it computes a weighted sum $v_i$ of its inputs $x:j$ which is passed ...
5
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1answer
64 views

Name for a property in a brutally elementary presentation of a monad

For evil reasons of my own, I'm trying to give a presentation of a monad in primitive terms, assuming only the notion of a category. More honestly, I looked at this post and got intrigued by the ...
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0answers
17 views

Name for “relative difference”

If we want to express that two numbers $x, y$ are not so far away from each other absolutely we use the absolute value function $|\cdot|$ with $0 < \epsilon \ll 1$: $$|x-y| < \epsilon$$ ...
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2answers
27 views

How do we emphasize that $\displaystyle x\mapsto\frac{1}{f(x)-y}$ “makes sense” if we know $y\notin\text{im }f$?

Please take a look at the following function $$x\mapsto\frac{1}{f(x)-y}$$ where $f$ is "some other function". Suppose we know $y\notin\text{im }f$, i.e. the expression in the denominator "makes ...
2
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2answers
26 views

Some basic terms from finite group theory, normalising and centralising

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, ...
0
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1answer
16 views

Difference between $C_0^{\infty}(U)$ with support in $A$ and $C_0^{\infty}(A)$

Let $A \subseteq U$ be open sets of $\mathbb R^n$. Is it true that $$ \lbrace f \in C_0^{\infty}(A) \rbrace = \lbrace f \in C_0^{\infty}(U) : \text{support of } f \subseteq A \rbrace \quad ? $$ I ...
1
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1answer
23 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
1
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2answers
42 views

Fundamental confusion on set theory and permutations

I am confused on the following: A set does not have any order. Now I read that a permutation is a bijection of a set. But doesn't this imply an order? I mean a bijection is a one-to-one function from ...
6
votes
1answer
54 views

Closed and Open Set - History of Terms

I know very little in the way of math history, but I question that was bothering me recently is where the terms open and closed came from in topology. I know that it's easy to ascribe a sense of ...
3
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1answer
48 views

Sets that are equal to closure of its interior

Is there a standard name for set $M$ for which $M = \overline{M^0}$? $M^0$ is interior and $\overline{M}$ is closure. Often I work with well behaved sets and functions. I have in mind continuous ...
0
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1answer
14 views

Is there any special name for an algebraic structure (set, equivalence relation)?

I've seen the term "real multiset" but it doesn't seem to be very appropriate so i wonder whether there are any others. My second question is about multisets. In most sources i've seen one of the ...
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0answers
46 views

Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...
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0answers
22 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
4
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2answers
402 views

Use of the word “solve”?

This is not a mathematical question, but just a matter of terminology. I don't understand why so many people (especially on MSE) want to solve integrals. It makes sense for me (linguistically ...
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1answer
37 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
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2answers
27 views

What's this equation called (one more each iteration, find total for given iteration)?

Say you have +1 on first iteration, +2 on second, and so on until N, and you want to know the total. That's easily calculate using (N * (N + 1) ) / 2. What's that equation or technique called?